ON THE ASYMPTOTIC ANALYSIS OF THE PAINLEVE EQUATIONS VIA THE ISOMONODROMY METHOD

During the last ten years considerable progress in the theory of classical Painleve equations has been achieved. This progress is based on the so-called Isomonodromy Method which makes the Painleve transcendents as effective in modern nonlinear analysis as the usual special functions are in linear analysis. In this paper, the problem of the rigorous justification of the global asymptotic results which are obtained via Isomonodromy Method, is considered. Taking the pure imaginary solutions of the second Painleve equation as an example, we discuss in detail three different rigorous methodologies of their asymptotic analysis including derivation of the corresponding connection formulae.